APM466-Assignment 1 Yield Curves Solved

2          Questions
2.1         Fundamental Questions
1.    (One sentence each.)

(a)     Why do governments issue bonds and not simply print more money?

(b)    Give a hypothetical example of why the long-term part of a yield curve might flatten.

(c)      Explain what quantitative easing is and how the (US) Fed has employed this since the beginning of the COVID-19 pandemic.

2.    We asked you to pull data for 32 bonds, but if you’d like to construct a yield a “0-5 year” yield & spot curves, as the government of Canada issues all of its bonds with a semi-annual coupon, when bootstrapping you’ll only need 10 or 11 bonds to perform this task. Ideally, the bonds in any yield curve should be consistent in some way with one another so that yields are easier to compare. Select (list) 10 bonds that you will use to construct the aforementioned curves with an explanation of why you selected those 10 bonds based on the characteristics we asked you to collect for each bond (coupon, issue date, maturity date, etc.).

Note: 1) There is a unique ideal answer, 2) To easily refer to a bond, please use the following convention: “CAN 2.5 Jun 24” refers to the Canadian Government bond with a maturity in June 24 and a coupon of 2.5.

3.    In a few plain English sentences, in general, if we have several stochastic processes for which each process represents a unique point along a stochastic curve (assume points/processes are evenly distributed along the curve), what do the eigenvalues and eigenvectors associated with the covariance matrix of those stochastic processes tell us?

(Hint: This is called Principal Component Analysis)

4.   Empirical Questions -
(a)     First, calculate each of your 10 selected bonds’ yield (ytm). Then provide a welllabeled plot with a 5-year yield curve (ytm curve) corresponding to each day of data (Jan 2 to Jan 15) superimposed on-top of each other. You may use any interpolation technique you deem appropriate provided you include a reasonable explanation for the technique used.

(b)    Write a pseudo-code (explanation of an algorithm) for how you would derive the spot curve with terms ranging from 1-5 years from your chosen bonds in part 2. (Please recall the day convention simplifications provided in part 2 as well.) Then provide a well-labeled plot with a 5-year spot curve corresponding to each day of data superimposed on-top of each other.

(c)     Write a pseudo-code for how you would derive the 1-year forward curve with terms ranging from 2-5 years from your chosen bonds in part 2 (I.e., a curve with the first point being the 1yr-1yr forward rate and the last point being the 1yr-4yr rate). Then provide a well-labeled plot with a forward curve corresponding to each day of data superimposed on-top of each other.

5.    Calculate two covariance matrices for the time series of daily log-returns of yield, and forward rates (no spot rates). In other words, first calculate the covariance matrix of the random variables Xi, for i = 1,...,5, where each random variable Xi has a time series Xi,j given by:

                                                                          Xi,j = log(ri,j+1/ri,j),              j = 1,...,9

then do the same for the following forward rates - the 1yr-1yr, 1yr-2yr, 1yr-3yr, 1yr-4yr.

6.    Calculate the eigenvalues and eigenvectors of both covariance matrices, and in one sentence, explain what the first (in terms of size) eigenvalue and its associated eigenvector imply.